2007 NJC CT Q10

In a chemical plant, a tank initially contains \(5000\) litres of water dissolved with a certain amount of substance \(X\). Solution containing \(0.05\)kg of substance \(X\) per litre of water flows into the tank at a constant rate of \(10\) litres per minute. The mixture is thoroughly stirred and the resulting solution flows out of the tank at the same rate such that the volume of liquid in the tank is kept constant. If \(x\)(kg) is the amount of the substance \(X\) in the tank at time, \(t\) (min), show that the rate of change of the amount of substance \(X\) in the tank can be modelled by the differential equation \(\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{250-x}{500}\).[2]

1. Solve the differential equation, leaving your answer in the form \(x=B-A{{\mathrm{e}}^{-\,Ct}}\) where \(A\) is an arbitrary constant, and \(B\) and \(C\) are positive constants to be determined.[3]
2. Deduce the long-term steady state amount of substance \(X\) in the tank?[1]